Displaying similar documents to “An F. and M. Riesz theorem for bounded symmetric domains”

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

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Suppose that μ is a Radon measure on d , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces p q s ( μ ) for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

Mean values and associated measures of δ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let u be a δ -subharmonic function with associated measure μ , and let v be a superharmonic function with associated measure ν , on an open set E . For any closed ball B ( x , r ) , of centre x and radius r , contained in E , let ( u , x , r ) denote the mean value of u over the surface of the ball. We prove that the upper and lower limits as s , t 0 with 0 < s < t of the quotient ( ( u , x , s ) - ( u , x , t ) ) / ( ( v , x , s ) - ( v , x , t ) ) , lie between the upper and lower limits as r 0 + of the quotient μ ( B ( x , r ) ) / ν ( B ( x , r ) ) . This enables us to use some well-known measure-theoretic results to prove new variants...

When is a Riesz distribution a complex measure?

Alan D. Sokal (2011)

Bulletin de la Société Mathématique de France

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Let α be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by α . I give an elementary proof of the necessary and sufficient condition for α to be a locally finite complex measure (= complex Radon measure).

An inversion formula and a note on the Riesz kernel

Andrejs Dunkels (1976)

Annales de l'institut Fourier

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For potentials U K T = K * T , where K and T are certain Schwartz distributions, an inversion formula for T is derived. Convolutions and Fourier transforms of distributions in ( D L ' p ) -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order α , 0 &lt; α &lt; m , of a compact subset E of R m has the following property: its restriction to the interior of E is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

L p boundedness of Riesz transforms for orthogonal polynomials in a general context

Liliana Forzani, Emanuela Sasso, Roberto Scotto (2015)

Studia Mathematica

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Nowak and Stempak (2006) proposed a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions, and proved their boundedness on L². Following them, we give easy to check sufficient conditions for their boundedness on L p , 1 < p < ∞. We also discuss the symmetrized version of these transforms.

Remark on the inequality of F. Riesz

W. Łenski (2005)

Banach Center Publications

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We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions | φ | p with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let C α be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density ̲ ( C α , E , r ) = i n f x C α ( E B ( x , r ) ) / C α ( B ( x , r ) ) for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that l i m r ̲ ( C α , E , r ) is either 0 or 1; the first case occurs if and only if ̲ ( C α , E , r ) is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let p = m k k = 0 p - 1 be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements μ N V N ( N ) , where V N are p N -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of p and N V N ¯ = L ( ) . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Iterates and the boundary behavior of the Berezin transform

Jonathan Arazy, Miroslav Engliš (2001)

Annales de l’institut Fourier

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Let μ be a measure on a domain Ω in n such that the Bergman space of holomorphic functions in L 2 ( Ω , μ ) possesses a reproducing kernel K ( x , y ) and K ( x , x ) &gt; 0 x Ω . The Berezin transform associated to μ is the integral operator B f ( y ) = K ( y , y ) - 1 Ω f ( x ) | K ( x , y ) | 2 d μ ( x ) . The number B f ( y ) can be interpreted as a certain mean value of f around y , and functions satisfying B f = f as functions having a certain mean-value property. In this paper we investigate the boundary behavior of B f , the existence of functions f satisfying...